The problem also elaborates that $x$ is a multiplicative inverse of $m\pmod n$, as is $y$.
All I've got so far is that, according to Bézout's identity, if $m$ and $n$ are coprime, $\exists x,y|mx+ny=d$, where $d=\gcd(m,n)=1$. This can be expanded using the Chinese remainder theorem, which says that $\exists x,y|mx\equiv 1\pmod n, ny\equiv 1\pmod m$.
From there I'm not exactly sure how to expand the proof any further. Any help would be appreciated.